p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.695C23, C4.1172- (1+4), (C8×Q8)⋊3C2, (C2×Q8)⋊7C8, Q8.8(C2×C8), C2.8(C23×C8), Q8○2(C22⋊C8), (C4×Q8).30C4, (C4×C8).30C22, C4.20(C22×C8), C4⋊C8.376C22, C42.227(C2×C4), (C2×C8).482C23, (C2×C4).679C24, (C22×Q8).32C4, C2.4(Q8○M4(2)), C22.44(C23×C4), C22.15(C22×C8), (C4×Q8).333C22, C22⋊C8.245C22, (C2×C42).786C22, C23.230(C22×C4), C42.12C4.32C2, (C22×C4).1283C23, C2.3(C23.32C23), C22⋊C8○(C4×Q8), (C2×C4⋊C4).79C4, (C2×C4×Q8).45C2, (C2×C4).32(C2×C8), C4⋊C4.251(C2×C4), (C2×Q8).229(C2×C4), (C2×C4).475(C22×C4), (C22×C4).140(C2×C4), SmallGroup(128,1714)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 228 in 198 conjugacy classes, 174 normal (12 characteristic)
C1, C2 [×3], C2 [×2], C4 [×14], C4 [×7], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×24], C2×C4 [×5], Q8 [×16], C23, C42 [×12], C4⋊C4 [×12], C2×C8 [×8], C22×C4, C22×C4 [×6], C2×Q8 [×12], C4×C8 [×12], C22⋊C8 [×4], C4⋊C8 [×12], C2×C42 [×3], C2×C4⋊C4 [×3], C4×Q8 [×8], C22×Q8, C42.12C4 [×6], C8×Q8 [×8], C2×C4×Q8, C42.695C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], C22×C4 [×14], C24, C22×C8 [×14], C23×C4, 2- (1+4) [×2], C23.32C23, C23×C8, Q8○M4(2), C42.695C23
Generators and relations
G = < a,b,c,d,e | a4=b4=1, c2=b, d2=e2=a2, ab=ba, ac=ca, dad-1=a-1, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=a2c, ede-1=a2d >
►On 64 points
(1 35 55 41)(2 36 56 42)(3 37 49 43)(4 38 50 44)(5 39 51 45)(6 40 52 46)(7 33 53 47)(8 34 54 48)(9 32 64 20)(10 25 57 21)(11 26 58 22)(12 27 59 23)(13 28 60 24)(14 29 61 17)(15 30 62 18)(16 31 63 19)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 31 55 19)(2 32 56 20)(3 25 49 21)(4 26 50 22)(5 27 51 23)(6 28 52 24)(7 29 53 17)(8 30 54 18)(9 42 64 36)(10 43 57 37)(11 44 58 38)(12 45 59 39)(13 46 60 40)(14 47 61 33)(15 48 62 34)(16 41 63 35)
(1 41 55 35)(2 36 56 42)(3 43 49 37)(4 38 50 44)(5 45 51 39)(6 40 52 46)(7 47 53 33)(8 34 54 48)(9 32 64 20)(10 21 57 25)(11 26 58 22)(12 23 59 27)(13 28 60 24)(14 17 61 29)(15 30 62 18)(16 19 63 31)
G:=sub<Sym(64)| (1,35,55,41)(2,36,56,42)(3,37,49,43)(4,38,50,44)(5,39,51,45)(6,40,52,46)(7,33,53,47)(8,34,54,48)(9,32,64,20)(10,25,57,21)(11,26,58,22)(12,27,59,23)(13,28,60,24)(14,29,61,17)(15,30,62,18)(16,31,63,19), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,31,55,19)(2,32,56,20)(3,25,49,21)(4,26,50,22)(5,27,51,23)(6,28,52,24)(7,29,53,17)(8,30,54,18)(9,42,64,36)(10,43,57,37)(11,44,58,38)(12,45,59,39)(13,46,60,40)(14,47,61,33)(15,48,62,34)(16,41,63,35), (1,41,55,35)(2,36,56,42)(3,43,49,37)(4,38,50,44)(5,45,51,39)(6,40,52,46)(7,47,53,33)(8,34,54,48)(9,32,64,20)(10,21,57,25)(11,26,58,22)(12,23,59,27)(13,28,60,24)(14,17,61,29)(15,30,62,18)(16,19,63,31)>;
G:=Group( (1,35,55,41)(2,36,56,42)(3,37,49,43)(4,38,50,44)(5,39,51,45)(6,40,52,46)(7,33,53,47)(8,34,54,48)(9,32,64,20)(10,25,57,21)(11,26,58,22)(12,27,59,23)(13,28,60,24)(14,29,61,17)(15,30,62,18)(16,31,63,19), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,31,55,19)(2,32,56,20)(3,25,49,21)(4,26,50,22)(5,27,51,23)(6,28,52,24)(7,29,53,17)(8,30,54,18)(9,42,64,36)(10,43,57,37)(11,44,58,38)(12,45,59,39)(13,46,60,40)(14,47,61,33)(15,48,62,34)(16,41,63,35), (1,41,55,35)(2,36,56,42)(3,43,49,37)(4,38,50,44)(5,45,51,39)(6,40,52,46)(7,47,53,33)(8,34,54,48)(9,32,64,20)(10,21,57,25)(11,26,58,22)(12,23,59,27)(13,28,60,24)(14,17,61,29)(15,30,62,18)(16,19,63,31) );
G=PermutationGroup([(1,35,55,41),(2,36,56,42),(3,37,49,43),(4,38,50,44),(5,39,51,45),(6,40,52,46),(7,33,53,47),(8,34,54,48),(9,32,64,20),(10,25,57,21),(11,26,58,22),(12,27,59,23),(13,28,60,24),(14,29,61,17),(15,30,62,18),(16,31,63,19)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,31,55,19),(2,32,56,20),(3,25,49,21),(4,26,50,22),(5,27,51,23),(6,28,52,24),(7,29,53,17),(8,30,54,18),(9,42,64,36),(10,43,57,37),(11,44,58,38),(12,45,59,39),(13,46,60,40),(14,47,61,33),(15,48,62,34),(16,41,63,35)], [(1,41,55,35),(2,36,56,42),(3,43,49,37),(4,38,50,44),(5,45,51,39),(6,40,52,46),(7,47,53,33),(8,34,54,48),(9,32,64,20),(10,21,57,25),(11,26,58,22),(12,23,59,27),(13,28,60,24),(14,17,61,29),(15,30,62,18),(16,19,63,31)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 5 | 5 | 0 | 0 |
| 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 5 | 12 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 15 | 0 |
| 0 | 0 | 2 | 0 | 15 |
| 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 15 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 12 | 5 | 0 | 0 |
| 0 | 7 | 7 | 5 | 5 |
| 0 | 7 | 10 | 5 | 12 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,5,5,0,0,0,5,12,0,0,0,0,0,5,5,0,0,0,5,12],[13,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[9,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,15,0,15,0,0,0,15,0,15],[16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,1,0],[1,0,0,0,0,0,12,12,7,7,0,12,5,7,10,0,0,0,5,5,0,0,0,5,12] >;
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4AD | 8A | ··· | 8AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | 2- (1+4) | Q8○M4(2) |
| kernel | C42.695C23 | C42.12C4 | C8×Q8 | C2×C4×Q8 | C2×C4⋊C4 | C4×Q8 | C22×Q8 | C2×Q8 | C4 | C2 |
| # reps | 1 | 6 | 8 | 1 | 6 | 8 | 2 | 32 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{695}C_2^3 % in TeX
G:=Group("C4^2.695C2^3"); // GroupNames label
G:=SmallGroup(128,1714);
// by ID
G=gap.SmallGroup(128,1714);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,219,100,675,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=b,d^2=e^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^2*c,e*d*e^-1=a^2*d>;
// generators/relations